Determinant approach of the $ (p, q) $-Hermite-Appell polynomials and some of their components

In this work, we offer the novel class of $ ({p}, {q}) $-Hermite-Appell polynomials. Some attributes of this class are constructed, along with the generating function, series definition, $ ({p}, {q}) $-derivative properties, $ ({p}, {q}) $-integral representation, summation formulas, and determinate representation. Additionally, we consider a few components for the $ ({p}, {q}) $ -Hermite-Appell polynomials and infer certain elements of their traits. The generating function and series expansions of some classes of two-dimensional $ ({p}, {q}) $-Hermite-Appell polynomials are provided. Moreover, we acquire a $ ({p}, {q}) $-differential operator formula for $ ({p}, {q}) $-Hermite-Appell polynomials. Finally, the Wolfram Mathematica software is used to plot the graphical diagrams of select components of $ ({p}, {q}) $-Hermite-Appell, along with two-dimensional $ ({p}, {q}) $-Hermite-Appell polynomials.